We have analyzed, using linear theory in the Boussinesq approximation, the stability of an unbounded sheared flow with system rotation around the spanwise (x2) axis and vertical (x3) stratification. The base flow is a particular solution of the Euler equations and the base Ertel potential is zero. The conservation of absolute potential vorticity —not new— is considered, and gives here an invariant of the motion for the disturbance field, in which the mean shear vorticity is called into play, in addition to system vorticity: a new important result. Conditions for validity of linearization of the equations for the perturbations were discussed. Plane-wave disturbances with time-dependent wave vector were considered and their governing equations were derived in a local frame attached to the wave vector and in which the incompressibility constraint is satisfied by construction. A linear differential system (with time-dependent coefficients) for the poloidal, toroidal and potential modes is derived. Due to the conservation of the Ertel potential, which leads to a constant of motion relating the toroidal and the potential modes, the rank-three differential system reduces to a rank-two one. In order to consider arbitrary initial conditions and solenoidal property for the velocity field, a reduced Green function g has been introduced.
An alternative formulation of the rank-two differential system of equations yields a non-homogeneous second-order differential equation with time-dependent coefficients. An ana-lytical solution has been derived for any orientation of the wave vector and for all values of
the rotation R and Richardson Ri numbers. The solution indicates that, for certain cases, all the components of the matrix g have not the same long-time behavior, so that an anal-ysis in term of exponential growth, as in normal-mode stability analanal-ysis, is necessary. This analysis relies only on the invariants on the Green function matrix. It is shown that one of the eigenvalues of the matrix g is unity, reflecting the existence of the linearized Ertel invariant, so that the sum of the two other ones is trace g−1 and their product is K/k, where k is the radial wavenumber at time t and K is its initial value. The determinant of g isK/k for sheared flows, even for time-dependent sheared flows [10] and with additional effect of rotation and stratification. Therefore the normal-mode stability analysis lies on trace g as the unique unknown to be calculated, and its long-time behavior was analyzed in terms of the rotation number, of the Richardson number, and of the azimutal angle ϕ.
It is shown that the flow stability is governed by the parameter σϕ2 =R(1 +R) sin2ϕ+Ri. The modes for which σ2ϕ >0 are stable, those for which σϕ2 = 0 are neutral, while those for which σϕ2 < 0 are unstable. In their linear stability analysis of accretion disks, Johnson &
Gammie [2] have only considered the case where ϕ = 0 (i.e., the k2 = 0 mode) for which rotation effects vanish.
It is shown now that, when σϕ2 < 0 (unstable case), the particular value ϕ = π/2 (i.e., the k1 = 0 mode) corresponds to the most unstable mode for which the solution exhibits an exponential growth, while the other unstable modes are characterized by an algebraic growth. An algebraic growth is found for the neutral modes (i.e., those characterized by σϕ2 = 0).The stable modes (i.e., those characterized byσϕ2 >0) undergoe a power law decay if 0< σϕ2 <(cos2ϕ)/4 or exhibit a damped oscillatory behavior ifσϕ2 >(cos2ϕ)/4.
For geophysical and astrophysical applications, stability diagrams have been plotted for all values of the Richardson and rotation numbers. For instance, in the (R(1 +R), Ri) plane, five domains have been distinguished. In the Ri >1/4 domain (i.e., the domain (I) in figure 2), the modes are stable exhibiting a damped oscillatory behavior. In the domain corresponding toRi <0 andR(1+R)<0 orR(R+1)<0< R(R+1)+Riand 0< Ri <1/4 (i.e., the domain (II) in figure 2), the modes are also stable but there are ones exhibiting a damped oscillatory behavior while the other undergo a power law decay. In the Ri < 0 and R(R+ 1)<0 domain (i.e., the domain (V) in figure 2), the modes are unstable. In the other two domains (i.e., the domains (III) and (IV), see figure 2) stable and unstable modes can coexist.
Because previous DNS studies for sheared homogeneous turbulence at high initial shear rate show that the linear theory contains the essential mechanism responsible for devel-opment of turbulence structures, and the region near the k1 = 0 mode has an important implication on the dynamics of large scales motion, the formalism used for the RDT-like sta-bility analysis is systhematically applied to compute the evolution of particular turbulence statistics: the streamwise “two-dimensional” kinetic and potential energies. It is shown that the long-time limit of the ratio, denoted by η∞(1) of these two-dimensional energies (potential over kinetic) depends on both rotation and stratification. As a preliminar evaluation of the relevance of the computation of the streamwise two-dimensional energy components, the variation of η∞(1) versus R or Ri for fixed values of σ = R(1 +R) +Ri has been compared to the long-time limit of that one of its three-dimensional counterpart, denoted by η, com-puted numerically. The conclusion that can be drawn from this comparison is that, when the couple (R(R(1 +R), Ri) lies in the domain (III) of the diagram shown in figure 2, the contribution coming from the k1 = 0 is not a dominant one in the evolution of energies. At least, this point shows the usefulness of a complete stability analysis as the one presentend in this study.
On the other hand, because in many accretion disks angular momentum is likely redis-tributed internally by MHD turbulence driven by magnetorotational instability (see Balbus
& Hawley [31]), it appears very useful for astrophysical applications to study, in a subsequent paper, the effects of a magnetic field on stratified rotating sheared turbulence. Finally, this linear study may pave the way for a fully nonlinear one using pseudo-spectral DNS in terms of comoving deformed coordinates [32].
Appendix
Equations for the poloidal, toroidal and potential modes
By using the expression of the pressure mode ˆpgiven by equation (15), the equations for the modes ˆui and ˆb given by system (16) can be rewritten as
duˆi
dt +Lijuˆj =
δi3−kik3
k2
ˆb, d
dtˆb=−N2uˆ3, kiuˆi = 0, (70)
with
Lij =S
δi1−2kik1
k2
δj3+ 2Ω
δin− kikn
k2
ǫn2j. (71)
The substitution of the relation ˆui = u(α)e(α)i where (i = 1,2,3), (α = 1,2), into the first equation in (70) implies
du(β)
dt +u(α)
"
e(β)i de(α)i dt
#
+e(β)i (Sδi1δj3+ 2Ωǫi2j)e(α)j u(α) = ˆbe(β)i δi3.
Because the local frame (e(1),e(2),e(3)) is an orthonormal one ande(1) is time-independent, then "
e(β)i de(α)i dt
#
= 0, (α, β = 1,2), (i= 1,2,3).
Therefore, the equations for u(1), u(2) and u(3) =−ˆb/N can be rewritten as du(β)
dt +h
e(β)i (Sδi1δj3+ 2Ωǫi2j)e(α)j i
u(α)+h
Ne(β)i δi3i
u(3) = 0, du(3)
dt −h
Ne(β)i δi3
i
u(β)= 0, (α, β = 1,2), (i= 1,2,3), that are equivalent to system (20).
Solution for the reduced Green function
When σϕ2 6= 0, we consider the second-order differential equation for g3i (see equation (38)) which has the particular solutiongp3i given by equation (41). To resolve the associated homogeneous equation we distinguish the case where k1 = 0 (i.e., ϕ = π/2 and the case where k1 6= 0.
Case where k1 = 0
When ξ2 =σ(k2h/k2)>0 withk2/k >0, the solution of equation (38) is found as g11 = 1− R(1 +R)
σ (1−cosξτ), g12= (1 +R)
√σ sinξτ, g13= Ri(R+ 1)
σ (1−cosξτ), g21 =− R
√σsinξτ, g22 = cosξτ, g23= Ri
√σ sinξτ, g31= R
σ (1−cosξτ)
g32=− 1
√σ sinξτ, g33 = 1− Ri
σ (1−cosξτ) (72)
Recall that σ = R(1 +R) +Ri. The solution corresponding to ξ2 = σkh2/k2 < 0, can be deduced from the above one by using the basic functional relations sinhx=−ısin (ıx) and coshx = cos (ıx). In that case, the solution exhibit an exponential growth. As for the solution associated to the case where σ = R(1 +R) +Ri = 0, it can also be deduced from the later solution,
g11 = 1−R(1 +R)k22 k2
τ2
2 , g12= (1 +R)k2
k τ, g13=Ri(1 +R)k22
k2 τ2
2 , g21 =−Rk2
k τ, g22 = 1 g23 =Rk2
k τ, g31 =Rk22 k2
τ2 2 , g32 =−Rk2
k τ, g33 = 1 +R(1 +R)k22 k2
τ2
2 . (73)
Therefore, at σ = 0, the solution exhibits an algebraic growth.
Case where k1 6= 0.
Atk1 6= 0,the homogeneous equation associated to (38) is tansformed (by using the pure imaginary variable z = ık3/kh) to equation (43) with solution g3j = C0jPµ(z) +C1jQµ(z).
So that, the solution of the non-homogeneous equation (38) can be written
g3j =C0jPµ(z) +C1jQµ(z) +gp3j (j = 1,2,3), (74) and, in view of equation (32) that can be rewritten as
g2j =
−ı√
1−z2cosϕdg3j dz , we obtain
g2j = (−ıcosϕ)√
1−z2
C0jPµ′(z) +C1jQ′µ(z)
, (75)
while, equation (29) allows us to determine g1j,
g1j =δ1j+ (Rsinϕ) (δ3j −g3j), g1j =δ1j + (Rsinϕ)
δ3j −C0jPµ(z)−C1jQµ(z)−g3jp
. (76)
Here, g3jp is given by equation (41), and C0j and C1j are constants that can be determined by using the initial conditions, gij(z0) =δij.The result is
C0j = 1−z02
"
δ3j −g3jp
Q′µ(z0) + ı cosϕ
Qµ(z0) p1−z02δ2j
# ,
C1j =− 1−z02
"
ı cosϕ
Pµ(z0)
p1−z02δ2j+ δ3j −g3jp
Pµ′(z0)
# . It follows that
g11= 1−g33p
1− 1−z20 Q′µ(z0)Pµ(z)−Pµ′(z0)Qµ(z) , g33 = 1−z20 Q′µ(z0)Pµ(z)−Pµ′(z0)Qµ(z)
+
1− 1−z02 Q′µ(z0)Pµ(z)−Pµ′(z0)Qµ(z) g33p , g22= 1−z20
1−z2 1−z20
1/2
Pµ(z0)Q′µ(z)−Qµ(z0)Pµ′(z) . Therefore, trace g is independent on the particular solution gpij.
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1
2 3
Shear
Vertical Stratification Rotation
Ω
3
S
B
FIG. 1: Rotating stratified shear flow: Ui = Sx3δi1 (Shear), Ωi = Ωδi2 (Rotation), B = N2x3 (Vertical stratification).
Ri
R(R+1) O
(d1)
(d2) Stable
1/4
−1/4
(a)
Unstable
Stable (b)
FIG. 2: Stability diagram in the (R(R+ 1), Ri) plane for given values of ϕn (see Eq. (55)) and ϕrs< ϕn (see Eq. (56)).
(d1): Ri =−R(R+ 1) sin2ϕrs+ (cos2ϕrs)/4.(d2): Ri =−R(R+ 1) sin2ϕn.
Stable (a): damped oscillatory behavior,|λ2|2 ∼e−y[A+r0cos (ℑµy+β)] withy= log(τcosϕ).
Stable (b): power law decay, |λ2| ∼(τcosϕ)µ with−1/2< µ <0.
Ri
R(R+1) O
1/4
−1/4
(I) (I)
(III) (V)
(V)
(IV) (II) (II)
FIG. 3: Stability diagram in the (R(R+ 1), Ri) plane.
(I): Ri >1/4.Region with stable modes exhibiting a damped oscillatory behavior.
(II): 0< Ri <1/4 andR(R+ 1)>0 or R(R+ 1)<0< R(R+ 1) +Ri and 0< Ri<1/4.Region with stable modes in which modes with damped oscillatory behavior and modes with power law decay can coexist.
(III):Ri<0< R(R+ 1) +Ri.Stable ((a),(b)) and unstable modes can coexist.
(IV): 0< Ri<1/4 andR(R+ 1)< R(R+ 1) +Ri<0.Stable (b) and unstable modes can coexist.
(V): Ri <0 and R(R+ 1) +Ri <0.Region with unstable modes.
ϕ
1 k2
k1 k2
ϕn
rs
Uns.
Stab. (b) Stab. (a) Stab. (a)
Stab. (b) k
Region (II) Region (III)
O O
(a) (b)
ϕrs
ϕ
1 k2
Stab. (b) k
O
n Uns.
Region (IV) (c)
FIG. 4: Stability diagram in the (k1, k2) plane.
(a) Region (II); (b) Region (III); (c) Region (IV) defined in figure 2. ϕn and ϕrs are defined by equations (55) and (56), respectively.
0
=4
=2 3=4
-2 -1.5 -1 -0.5 0 0.5 1
'
R=2=S Unstable Stable
(a)
Stable
(b)
Stable
(a)
Stable
(b) i
0
(b) R
i
= 0:1
=4
=2 3=4
-2 -1.5 -1 -0.5 0 0.5 1
'
R=2=S Unstable Stable
(a)
Stable
(a)
(b) (b)
(b) (b)
0.
() R
i
=0:
=4
=2 3=4
-2 -1.5 -1 -0.5 0 0.5 1
'
R=2=S Unstable Stable
(a)
Stable
(a)
(b) (b)
(b) (b)
FIG. 5: Stability diagram in the (R, ϕ) plane. ϕrs given by equation (56) is represented by doted lines. (a)Ri= 0.1,(b) Ri =−0.1, (c)Ri= 0.
-6 (equation (67)) and its 3-D counterpartη=Kp/Kversus the rotation numberRfor a fixed positive value of σ=R(1 +R) +Ri.(a) σ= 0.1,(b) σ= 1.
-10 (equations (68)-(69) ) and its 3-D counterpartη=Kp/K versus the rotation numberR for a fixed value of σ=R(1 +R) +Ri.(a) σ= 0.0,(b) σ=−0.1